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Chapter 9
Kiara, Chelsea, Angus
9.1 Basic Terms of the Circle
Circle
• Set of points in a plane at a given distance (radius)
from a given point (center)
• Nothing in interior is considered a circle
• 2D
Sphere
• Set of points in space w/ distance (radius) from a
given point
• 3D
Tangent
• A line in a plane of a circle or sphere that intersects
the circle/sphere in exactly one point
Chord
• A segment whose endpoints lie on a circle or a sphere
• A diameter is a chord that contains the center of the
circle
– Longest chord of a circle
Secant
• A line in a plane of a circle or sphere that intersects a
circle in 2 points
• A line that contains a chord
Congruent Circles
• Circles that have the same radii but not
necessarily the same center
Concentric Circles
• Circles that lie in the same plane and have the
same center but not the same radii
• Ex: rock thrown in water
Polygons and Circles
• A polygon is inscribed in a circle or the circle is
circumscribed about the polygon when each
vertex of the polygon lies on the circle
When 2 circles do not touch
• Can construct 4 common tangents
– 2 External Tangents
– 2 Internal Tangents
• Internal line of center of circle
When 2 circles touch externally
• Can construct 3 tangents
– 2 external tangents
– 1 internal tangent
When 2 circles intersect each other at 2 points
(overlap)
• Can construct 2 external tangents
When 2 circles touch internally
• Can construct only one tangent
– Intersects both
2 concurrent circles
• Can NOT construct any common tangent
– Zero
9.2 Tangents
Tangent Perpendicular Theorem
• If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the point of
tangency
• Angle <ATO = 90
Corollary
• Tangents to a circle from a point outside the circle are
congruent ≅
• Prove by H-L and CPCTC
C. Tangent Perpendicular Theorem
• If a line is perpendicular to the radius at its outer end
point, then the line is tangent to the circle
B
Name each of the following:
Two radii: BO, OD
A secant : AB
Two chords: AB, BD
A diameter : BD
A tangent: ED
A point of tangency: D
B and D are the centers of
the circles.
A
O
D
A
B
14
Find the length of AC
E
C
D
7
9-3 : Arcs and Central Angles
Central Angle
●
An angle with its vertex at the center of the circle
Arc
●
An unbroken part of the circle
Minor Arc
●
Points in the interior of central angle
●
Named by endpoints
Major Arc
●
2 points & remaining points of center circle
●
Named by 3 points
Semicircles
●
If 2 points are the endpoints of a diameter then 2 arcs are called
semicircles
Measure of a Semicircle
●
M = 180
Measure of a minor angle = The measure of its central angle
Measure of a Major Arc
●
360 – measure of minor arc
Adjacent Arcs
●
Arcs that have exactly one point in common
Arc Addition Postulate
●
The measure of the arc formed by 2 adjacent arcs is the sum of the
measures of these 2 arcs
Congruent Arcs
Arcs, in the same circle, or in congruent circles that have equal
measures
Th. 9.3
In the same circle or in congruent circles, two minor arcs are congruent if and
only if their central angles are ≅
Problems 9-3
Find measure of c ent ral angle 1
150
At 11 o'c loc k the hands of a clock form an
angle of ____ ?
1
T he hands of a c loc k form a 120 angle at ___
o'c loc k and at ___ o'c loc k.
mCB
60
M<1 ?
M<2 ?
70
?
?
?
56
?
?
?
25
?
?
x
9.4 Arcs and Chords:
•Theorem 9.4
–In the Same Circle or in congruent circles congruent arcs have congruent chords
–In the Same Circle or in congruent circles, congruent chords have congruent arcs
•Theorem 9.5
–A diameter that is perpendicular to a chord bisects the chord & its arc
•Theorem 9.6
–In the same circle or in congruent circles, chords equally distant from the center are congruent
–In the same circle or in congruent circles, congruent chords are equally distant from the center
Practice Problems 9-4
D
B
O
N
M
E
13
C
PQ = 24; OM = ____?
BC = 18; OM = 12
ON=10 ; DE = ___?
9.5 Circles – Inscribed Angles
Inscribed angle :
• Angle with vertex on circle and sides which are chords of the circle.
Intercepted Arc
• The points on the circle in the interior of an angle of the circle.
Theorem 9-7
• The measure of the inscribed angle is half as much as the intercepted arc
Corollary 1:
• If two inscribed angles intercept the same arc or congruent arcs, then the angles are
congruent
Example … Find X :
• Since X is an inscribed arc, then the measure of the angle is ½ the
measure of the arc
• So, x = 54
Try it! Find X:
Corollary 2:
• An angle inscribed in a semicircle is a right angle
Corollary 3:
• The opposite angles of an inscribed quadrilateral are supplementary
Corollary 4:
• The angle formed by a tangent and a chord equal 1/2 the measure of the intercepted arc
9.6 Other Angles
Angle Formed by 2 chords:
• Is the average of the 2 arcs
• M<X = (mBD + mCA) ÷ 2
Angle Formed by 2 secants
• Is the difference of the 2 arcs
• m<X = (mAB – mCD) ÷ 2
Angle formed by 2 tangents
•
•
Is (major arc – minor arc) ÷ 2
M<X = (mACB – mAB) ÷ 2
A
Angle formed by a secant & a tangent
•
M<X = (mAD – mAC) ÷ 2
Example Problems… Find X:
9.7 Circles & Lengths of Segments
Theorem 9.11
• RxQ=TxS
Theorem 9.12
• DE x ME = FE x NE
Theorem 9.13
• R x S = T2
• NR x KR = SR2
S
Q
R
T
S
R
T
• Try it! Find X: